My references (including Culp) and my intuition, like yours, suggest that dividend should be treated the same way. And the theory, too: specifically, the cash or nothing is a special case of BSM.

And the test, it seems, is to compute the cash-or-nothing call and the cash-or-nothing put. The combination of the positions--note the elegance of this(!)--must be $1; i.e., at strike = $1, the payoff is a certain $1.

And so, using your numbers:
cash-or-nothing call = $0.5101 (note this is a d2 that is influenced by d1; by my d1 = 0.1945 per the -div. Please note that subtracting the dividend outright is equivalent to the alternative of embedding it inside the LN(). That is: LN(S*EXP[-q*T]/K) = LN(S) + LN(EXP[-q*T]) - LN(K) = LN(S/K) + (-qT) = LN(S/K) - qT ... so just a reminder the "adjustment" can be found in one of two places and your texts may not explicitly refer to that....or perhaps they merely assume a non-dividend paying stock

In this case, I get:
cash-or-nothing call = $0.5101
cash or nothing put = $0.46878
and sum of cash or nothing put and call = $0.97897 (cell C31) which should equals the discounted value of a certain $1 to be received in 0.5 years(!) = EXP(-4.25%*0.5), which it does.
... so this would appear to validate your intuition that the treatment of dividends is the same